I
P. M. YAVORSKY, R. J. FRIEDRICH, and EVERETT GORlN Research and Development Division, Consolidation Coal Co., Library, Pa.
Heat Transfer and Thermal Stresses in
....
Carbonization of Briquets This article shows how a digital computer was used to develop data for a continuous process for the production of formcoke suitable for blast furnace use. The potential advantages of such a process are lower investment costs per ton of coke, ability to utilize the vast reserves of nonmetallurgical grade coal, and production of a product coke having uniform size and quality
THE
work reported is part of a program for development of a continuous process for the production of formcoke suitable for blast furnace use. Experimental work has shown that such a process, yielding strong intact formcoke, can be based on briquetting followed by continuous coking of the briquets. The briquets are formulated from low temperature char, coal, and pitch binder by commercial briquetting machinery. Shock heating is necessary to prevent plastic deformation during carbonization, but too severe heating will lead to fracturing. This article is basically addressed to the computation of the heating rates and temperature distribution in briquets during carbonization by various methods of shock heating, thus supplying process design data. A comparison is made where possible with heating rates determined experimentally. Another objective is to determine from the calculated thermal patterns the relative magnitudes of the thermal stresses, which in excess are believed responsible for deleterious fracturing. Procedure The problem of heat conduction in
carbonaceous materials has been attacked previously. Burke and others ( 2 ) some years ago discussed mathematical relationships for the rate of heat conduction through coal undergoing carbonization. However, to arrive at an analytical solution, they treated the case that corresponded to the assumption that the thermal diffusivity, a, is independent of temperature. A constant 01 for carbonaceous materials cannot be assumed. The strong temperature dependence of thermal conductivity, k, and specific heat, G, has been shown by experimental measurements ( I ) . a is given by CY = k/pc, where p is the density. The nonconstant conductivity of coal was also reported by Millard ( 6 ) , who attempted to solve the Fourier heat-flow equation by an electrical analog technique. H e could obtain agreement between calculated data and experimental thermal patterns in a coke oven only by injecting sizable heats of carbonization into the calculations. The problem can be handled in principle by solution of the Fourier heat conduction equation. T h e equation, in polar coordinates, for the general case of a sphere wherein the thermal diffusivity is temperature dependent, is
This equation must be solved with boundary conditions T = 7'1 at t = 0 for all values of r and k
dT = h(T0 dr
-
T I )at
1
=
YO
(2)
To, the ambient temperature of the heating medium surrounding the briquet, varies unless the heat content of the medium greatly exceeds that of the briquet. Its variation with time can be derived from heat balance, yielding
This relation holds for continuous carbonization where concurrent flow of heating medium and briquets prevails, and for batch carbonization where no temperature gradient exists within the heating medium. Thus heat flow Equation 1, with the compIex boundary conditions, as well as the nonconstant parameters k and c, cannot be solved by the usual analytical VOL. 51,
NO. 7
JULY 1959
833
1832
1832
CASE 1652
CASE
1652 T.
Ip
2-INCH BRIQUETS : 1800DF
1472
2-INCH BRIOUETS
1472
To = 2 6 0 0 p F M :091
hr,
= 0 079 B l u / h r f i 'F
1292
1292 1112 0
1112
w
a
$ a
w
932
a
w
t
a
E+
932
[L
752
w a
E 752 I-
572
572
392 392
212 212 O b
200
460
600 800 IO00 T I M E , SECONDS
1200
00
Figure 1. In shock heating with 1800' F. gas uniform temperature is approached in briquet after 1400 seconds
200
400
600 TIME
800
, SECONDS
1000
1200
Figure 2. In shock heating with 2600' F. gas temperature drops sharply Uniform temperature i s approached i i shorter time when higher gas temperature i s used
methods. A numerical method of solution is necessary. Although these equations cannot be solved analytically, certain interesting properties of the solutions follow from the form of Equation 1 and the boundary conditions. Let new variables be introducedX = r/ro> t' = t/ro2: and h' = hro. I t can now be verified that the temperatures, as expressed in Equations 1 and 2, become functions only of X , t', k , h', and M . It follows that the solution is a function only of X and t ' , if k , M , and hro are maintained constant. Consequently, an equivalent temperature profile is established within a briquet at corresponding values of the relative radius, X,at a time inversely proportional to the square of the radius of the briquet. This is a more generalized expression of the law of squares which has been discussed (6) and will become useful later. I t is necessary, before proceeding with the numerical analysis, to establish values for the parameters needed-for briquets of Pittsburgh Seam coal, product char, and binder pitch. Empirical equations, used in subsequent machine computations, were de834
rived giving the dependency of the specific heat and thelmal conductivity of the briquets with temperature. The data for specific heats for the coal and char as well as the thermal conductivity of the briquets were experimental (I). The specific heat of the binder pitch was taken from Hyman and Kay ( 4 ) . The measured apparent density of the raw briquets of 0.8 gram per cc. was used. I t was assumed not to change during carbonization. Evaluation of the film coefficients has been explained ( I ) . Values of 20 and 50 B.t.u.'hr. sq. ft. F. were assigned for the computations on solids heating, to bracket expected values for heat transfer from fluidized solids to briquets. For the gas film heat transfer coefficient. 9.5 B.t u.jhr. sq. ft. F. was employed in calculating the heating rate of 2-inch briquets by hot flue gas. This value derives from correlations by Gamson and others ( 3 ) and Wilke and Hougen (8) and corresponds to a flow rate of 375 pounds of gas,/sq. ft. hr. The problem of the rate of heating of briquets can be solved by application of numerical methods of analysis similar to the step methods described by Ingersoll
INDUSTRIAL AND ENGINEERING CHEMISTRY
O
(5). I t was solved by machine computation on an IBM 650 computer. T o facilitate numerical solution, the physical process of heat transfer is artificially resolved into two distinct, sequential processes: isothermal flow of heat from one section of matter to another for a short time, and a resultant change in temperature of the section based upon its heat balance during the isothermal period. I n reality, the flow of heat and change of temperature are not sequential, as pictured here, but occur simultaneously, so that solution of the problem is approximate, though of very close approximation. The numerical solution approaches rhe rigorous one as the increments of time and space that enter the computations are made smaller, for if the increments become infinitesimal, the solution will be a true calculus integration or the differential equations. The use of an electronic digital computer, with irs extremely rapid computation, permits computation of a set of equations for a great number of very small time and space increments ; thus, the solution is very nearly rigorous. I n this approximation method, a briquet is considered as made u p of ten
C A R B O N I Z A T I O N OF BRIQUETS
1472LcASE =
1472
1292
1292
1112 LL
932 W
K
3
t a
752
W
6'
n
z
f
T I M E , SECONDS
572
Figure 4. In shock heating of large briquets with char a long time of 2000 seconds is required to approach uniform temperature for large 3-inch briquet:
I- INCH BRIQUETS To l3SOoF M =SO hro = 2 08 Btu/hr ft O F t' : I612 (tlr:)
392
8
260 400 600 800 1000 I 2 0 0 1400 1600 I 8 0 0 2000 2200 2400
Note time l a g before center temperature starts to rise
212
4 Figure 3. 0
In shock heating with fluidized char smaller briquets approach uniform temperature very rapidly
0
100
20 0 T I M E , SECONDS
concentric spherical layers having initial temperatures of 1Tl, l T z .. . LTlo. The briquet is heated by hot gas of initial temperature 1To. For the duration of a short time interval, AT, these temperatures are assumed constant while heat conduction proceeds, the driving force being differences between 1 To, 1T I . 1Tlo. The heat transferred from the gas to the first layer, as well as between layers, is easily derived for the temporary steady state from Newton's law for heat exchange and the basic Fourier equation for radial heat transfer ( 5 ) . For the computations, these differential equations are written as finite increment or difference equations. The balance of heat left in any layer or in the heating medium at the end of a time interval is obtained as the difference between heat transferred into and out of layer or medium. Next, the temperatures of the gas and layers are allowed to change by virtue of the balance of heat lost or gained by applying heat capacity equations. All heat terms can then be eliminated, yielding as the final equation for computations
..
A,zTi = B,-i iTi-1 (Bi-1 B, - A < ) 1Ti
+
+ BilTi+l
(4)
where the A , and Bd terms are calculable constants for a particular time interval, A t , involving h, k(lTi), c ( l T J , A t , and briquet dimensions. Because the IT values are known from chosen initial conditions, the temperatures at the end of a time interval, zT,, are the only
300
Note
unknowns involved in Equation 4, which is completely general for the heating medium and each briquet layer. Thus, the temperature of the gas, and of each layer in the briquet, at the end of the first time interval, is computed directly from Equation 4. These end temperatures, 2Tz, then become the initial temperatures, ITs, for the next time interval, and the computations are repeated with new A, and Bi terms to determine the temperatures at the end of the second interval. Iteration of this procedure, for many time intervals, in the computer yields the temperature profile of the gas and the layers of the briquet as a function of time. Tensecond intervals were used in ccmputations for 2- and 3-inch briquets and 1second intervals for the 1-inch briquet.
Discussion Calculated Results. Eight briquet cases were solved directly on the computer covering variation of the parameters (see table). The law of squares can be used to apply the results to different size spheres by adjusting the value of h to maintain hro constant. For brevity of presentation, the calculated temperature distribution patterns are illustrated for only four of the cases in Figures 1 through 4. The time scale in each case is given for a 1-, 2-, or 3-inch briquet as noted in the figure. The conversion factor to convert the time scale to other briquet sizes by the law of squares is also noted.
small drop in char temperature
Two theoretical checks on the computed results lend some degree of confidence to the accuracy of the numerical method applied here. For every computed case for various conditions, the equilibrium temperature was near 1100" F., as predicted by over-all heat balance. This would probably not occur if there were a gross error in the numerical method. The computed results of the various cases obey the law of squares-that is, the computed time to reach a specified temperature varies exactly as the square of the radius, under the necessary condition that hr, is constant. This is easily tested from the data in Figure 5. Comparison with Experimental Results, Some experiments to compare the temperature rise at the briquet center with theoretical behavior were arranged so that the results could be compared with cases IV through VIII, where char was used as the heating medium. These computed cases were set up with a decreasing temperature of the char from an initial value of 1350 " F. to a final equilibrium temperature of lllOo F. This temperature pattern (set up as representative of a continuous process) could not be conveniently reproduced in the laboratory. The experiments were, therefore, conducted at a uniform 1200" F. in the 8inch fluidized sand bath. A thermocouple was inserted into the center of the briquet, which was plunged into the VOL. 51,
NO. 7
JULY 1959
835
sand bath, and the temperature history was continuously recorded. I t was assumed that the heat transfer properties of fluidized char and fluidized sand are not very different. Measurements of this kind were made with I-, 2-, and 3-inch diameter spherical briquets. The linear fluidizing velocity of the sand bath was maintained constant at 0.26 foot per second for 1- and 2-inch briquets. T o check the effect of linear velocity, a second and higher velocity of 0.45 foot per second was also employed for the 2-inch briquets. As no effect of fluidizing velocity was found in this range, 3-inch briquets were measured only at 0.45 foot per second. The experimental measurements are compared with the calculated rate of temperature rise in Figure 5 for all three briquet sizes. The curves shown are calculated for different values of h as parameter. The work reported for aluminum spheres (7) indicates that the correct value of h is in the neighborhood of 30 B.t.u./hr. sq. ft. In all cases the rate of temperature rise initially is greater than the calculated figures. That this phenomenon is undoubtedly mainly due to a conduction thermocouple error was shown by experiment. The exposed extrusion of a thermocouple injected into the center of a 2-inch briquet was heated electrically to l l O O o F. Another unheated couple was inserted also to the briquet center at a 90" angle to the former one. The heated thermocouple read as much as 100' F. above the unheated couple. This error is sufficient to bring about agreement between the lower temperature experimental points and a calculated curve expected for an h of 30 for 2-inch briquets (Figure 5). As the temperature of the briquet rises: the thermocouple error naturally would diminish rapidly because of the smaller temperature differential and the increase in thermal conductivity of the briquet material. In the case of the 2-inch briquet, the effect of fluidizing velocity- on the experimental rate of temperature rise is negligible. Good agreement between theory and experiment is obtained after the temperature rises above 300' C. by assigning a lower than predicted value of h, of the order of 20. The agreement at higher temperatures becomes rather poor in the case of the 3-inch briquet, because a value below 13 must be assigned to h to obtain agreement. This higher temperature discrepancy is thought to be due to the retarding effect of volatile matter release on the penetration of heat through the surface of the briquet-i.e., effectively on the value of h. This effect was neglected in the calculations because of the added complexity it would have introduced. However, the rate of volatile matter release per unit of briquet surface in-
836
FLUID VELC. f pm
EXPERIMENTAL POINT
I"
1112.
.O 0
0.26 0.26 0.45 0.45
BRIQUET 0 P' BRIOLET A 2" BRIQUET 3'' m a U E T
200
400
600 800 T I M E , SECONDS
1000
1200
1400
Figure 5. Experimental and calculated heating rates agree adequately up to to 2-inch size Larger briquets show slower heating rates than calculated
creases in proportion to the briquetradius, making it more serious for the larger briquets. Empirical Correlations for Heating Rates A simplified correlation encompassing the calculated results would simplify extrapolation to cases that were not directly considered and the application rate of heating to many design problems. A correlation was developed to fit the four cases, V through VIII, considered for solids heated briquets. These cases correspond, for a 2-inch spherical briquet, to a range of h values of 20 to 75 B.t.u./ hr. sq. ft. O F. The correlation is based on the use of the wholly empirical equation
where T is approximately the mean briquet temperature and T, is the temperature of the fluidized solids medium. T i s exactly defined by
-
T = TO+ Q / E
(6)
To is the initial briquet temperature, Q is the amount_ of heat absorbed by the briquet, and C is the mean specific heat of the briquet-over the whole carbonization range. T would be exactly equal to the mean briquet temperature if the specific heat of the briquet were independent of temperature, which it is not. The test for this equation is shown in
INDUSTRIAL AND ENGINEERING CHEMISTRY
r)
Figure 6, where log ( T 8is plotted against time. The points shown are derived from the computed results of cases V through VIII. In the cases studied here, heat balance considerations give rise to the relationship T, -
T
=
To
+ 5.73 - 1.23T
(7)
where TOis the initial temperature oC the heating medium in degrees centigrade. The slopes of the straight lines shown in Figure 6 are equal to K of Equation 5 multiplied by 1.23. It is clear that the correlation holds with adequate accuracy, as noted by the linearity of the plots. The calculated slopes K' (= 1.23 K ) are given for the h values on the graph. It now remains, to complete the correlation, to account for the variation of K with h dnd with briquet radius Y . The variation with h, at constant Y of 1 inch, is adequately expressed by the empirical equation
where a = 0.00834, 6 = 0.0369, the time is in minutes, and h has units of B.t.u./hr. sq. ft. ' F. Equation 8 is readily transposed to other briquet sizes by the law of squares. The final correlation is given below, where r is in feet and t in minutes. d-T -dt
C A R B O N I Z A T I O N OF BRIQUETS This equation fits the computer calculations shown in Figure 6 with an accuracy to 5%.
1472
Thermal Stresses Thermal stresses can arise during coking as a result of the temperature distribution produced within the briquet, combined with a contraction or expansion of the briquet material. For better understanding of the nature of the thermal stresses within the briquet, a brief dilatometer study was made of the thermal expansion and contraction characteristics of the briquet material. T h e studies were made on samples of briquets of material from Pittsburgh Seam coal, particularly from the ArkWright and Moundsville mines in West Virginia. The dilatometer is an electrically heated vessel about 1 cm. in inside diameter containing a sample 2 cm. high. The sample supports a rod, counterbalanced by a small weight attached to a string which passes over a pulley. The pulley is fitted with an indicating needle whose displacement can be calibrated in terms of linear expansion or contraction of the sample. At a slow rate of heating, about 3 O to 5" F. per minute, the sample is assumed to be at the same temperature as the container, and essentially uniform throughout. The data for briquets from the two coals are shown in Figure 7. The sharp contraction seen with the briquet is not observed with the loose mix. The compacted briquet mix remains in a plastic condition due to softening of the pitch and coal, until a rigid coke bond is formed. Therefore, thermal stresses can be set up only within the rigid coked crust of the briquet, while the inner plastic region undergoes relaxation of any imposed stresses by flow into the rigid shell. The problem can therefore be handled by application of Equation 10 for the case of thermal stresses in a hollow sphere. The thermal stresses are a function of the tempera-
Figure 6. Empirical correlation of rate of heating of briquets shows adequate agreement with calculated points
2
'
'
'
6'
'
8'
'
1'0
'
1I2
TIME, MINUTES
ture distribution within the body, the shape of the body, the coefficient of thermal expansion, a, the modulus of elasticity, E, and Poisson's number, y . For an isotropic and elastic body, the tangential stress, ut, at a radial position, Y, of a hollow sphere with an inner radius a and an outer radius b is given by Timoshenko (7) as follows
& l T r z a ' r - '/zT1 (10) This expression cannot be used to calculate the exact magnitude of the stresses existing in briquets during coking. The main difficulty is that the material is not homogeneous and is changing in chemical and physical structure with time and temperature. The briquets actually shrink rather than expand as the temperature rises. I n a rough approximation of the dilatometer results shown in Figure 7 (right), the shrinkage may be treated as linear with the temperature. Under such conditions the Timoshenko equation can be employed to calculate relative thermal stresses by treating the factor &/(1 - Y)
'
Mia HLATING
i: - 2 , W
.
212;.
0 .
P L
931 752
as an unknown parameter for different coking conditions, but assumed to be constant. Application of this method also requires a more or less arbitrary decision relative to the temperature at which plasticity of the mix disappears. I t has been assumed in what follows that the mix becomes rigid at 800' F. The method adopted, therefore, was to compute the relative thermal stress over the coked portion of the briqueti.e., over the shell where the temperature was 800° F. and higher as a function to time, coking conditions, and briquet size. For simplicity, the calculations were restricted to computing the tangential stress at the surface of the briquet only. Application of the Timoshenko equation to the calculation of thermal stresses can be criticized, because the equations apply to an elastic body of constant chemical structure, which is not the case here. However, in any case the thermal stresses are greater the sharper the temperature gradients within the body. The Timoshenko equation merely provides a convenient framework for a semiquantitative evaluation of relative thermal gradients for different heating patterns.
--HEATHO
x -2
0
-4.
r -4
-6.
c
9RlQUET COOLING
_ _ _ _ _ - - - - - - - - -_-_- - - - _ _ _-__
;-a. w -10
-I&?
0
Figure 7. briquets
200
400
600
800
1000
I200
1400
Sharp thermal contraction is observed between
-I4 0
cooLina 200
400
600
800
1000
1200
1400
1600
600" and 800° F. for Arkwright (left) and Moundsville (righf) VOL. 51,
NO. 7
JULY 1959
837
Nomenclature
I l l
INITIAL FLUID TEMPERATURE
,000
I200
Figure 8. Relative thermal stress for char-heated briquets increases with briquet size and decreases with film coefficient
T o establish a background of comparison for determining which relative stresses can be expected to exceed the fracturing limits, a regime has been worked out experimentally for shock carbonization of briquets without fracturing. Intact and nondeformed 2inch formcoke was produced by shock heating in a fluidized sand bath, if the sand temperature was within the range of 300 to 1150” F. The proper regime for hot gas formcoking is difficult, if not nearly impossible, to investigate on a laboratory scale. T h e thermal stress for char heated briquets as a function of briquet size, time, and film coefficient is shown in Figure 8. The equilibrium temperature in all cases was constant a t 1100” F. The relative thermal stress in 2-inch briquets heated with gas was also calculated by the above method, where again the equilibrium temperature was maintained near 1100” F. The thermal stress in the gas cases was lower than the char heating cases, even when 2600” F. gas was used. The major increase in maximum stress is between the 1- and 2-inch size, with a smaller increase between 2 and 3 inches. This complies with experimental findings, that 1-inch briquets survive the successful regime established for 2-inch briquets.
Case I I1 I11 IV V VI VI1
VIII
Experimentally, it is difficult to produce fracture-free 3-inch carbonized briquets by shock heating in fluidized sand. The increase in thermal stress in going from 2- to 3-inch briquets must be critical. Conclusions The calculational procedure is sufficiently precise to predict heat transfer rates and temperature distribution patterns adequately within spherical. shockheated briquets 2 inches or less in diameter. I t is less accurate for larger briquets because of the error resulting from neglect of the evolution of volatile matter. The method is extremely useful for the investigation of cases such as heating with hot gas, which cannot easily be investigated experimentally. For example, it predicts that 2-inch briquets may be successfully shock-heated using hot gas and without fracturing with gas a t temperatures above 2600” F.
1To = temperature of hot gas or hot fluidized solids at beginning of specified time interval To = ambient temperature of heating medium I Ti, i = 1 . . . n = temperatures of spherical layers in a briquet of n layers, at beginning of time interval. nth layer is center ZTO = temperature of hot gas or hot fluidized solids at end of specified time interval z T ~i ,= 1 . . . n = temperatures of briquet layers at end of specified time interval At = time interval, seconds r = radius in general Ar = thickness of briquet layer ro = radius of solid spherical briquet k ( T ) = thermal conductivity of briquet material, a function of temperature c( T ) = specific heat of briquet material, a function of temperature 12 = film heat transfer coefficient for transfer from gas or fluidized solids to briquet n = number of layers in briquet, arbitrary a = kjpc, thermal diffusivity. Also used for expansivity M = mass ratio of heating medium to briquet T = mean briquet temperature TI = initial briquet temperature literature Cited (1) Batchelor, J. D., Yavorsky, P. M., Gorin, Everett, J . Chem. Eng. Data 4, 248 (1959). (2) Burke, S. P., others, Am. Gas Assoc. PYOC. 1930, 700, 820. (3) Gamson, B. W., others, Trans. Am. Zmt. Chem. Engrs. 39, 1 (1943). (4) Hyman, D., Kay, W. B., TND. ENG. CHEM.41, 1764 (1949). (5) Ingersoll, L. R., “Heat Conduction,” University of Wisconsin Press, Madison, Wis., 1954, (6) Millard, D. J., J . Inst. Fuel 28, 345 (1955). (7) Timoshenko, S., “Theory of Elasticity,” McGraw-Hill, New York, 1934. (8) Wilke, C. R., Hougen, 0. A., Trans. Am. Inst. Chem. Engrs. 41,445 (1945).
Acknowledgment Thanks are extended to William Kehl and Nicholas Sabers, University of Pittsburgh Computing Center, for helpful discussions and the final detailed translation of the numerical analysis into computer language.
Formcoking Cases for Which Thermal Patterns Were Solved Mass Ratio, hro, h -+ B.t.u./”r. Sq. Ft. O F., Heating Temp., F. Medium B.t.u./Hr, 70 Medium Initial Equil. Briquet Ft. Sec. 1-Inch 2-Inch 3-Inch 19.0 9.5a 6.3 2200 705 0.58 0.079 Gas 1000 1.24 0.079 19.0 9.5” 6.3 2200 Gas 0,079 19.0 9.5“ 6.3 Gas 1800 1000 1.95 0.079 19.0 9.9 6.3 Gas 2600 1000 0.91 25.0 16.7 Char 1350 1110 5.0 2.08 50.Oa 4.16 100.0 50.0“ 33.3 Char 1350 1110 5.0 6.24 150.0 75.0 50.0Q Char 1350 1110 5.0 1.67 40.0 20.0a 13.3 Char 1350, 1110 5.0
RECEIVED for review November 12,, 1958 ACCEPTED April 17, 1959 Division of Gas and Fuel Chemistry, 134th Meeting, ACS, Chicago, Ill., September 19’jS.
3 Detailed mathematical derivations of Equation 4 are available from the authors on request. The companion article on “Heat Transfer and Thermal Stress in the Carbonization of Briquets” appears in the Julyissue (Vol. IV, No. 3) of the Joztrnal of Chemical and Engineering Data.
Case also fits for the other t w o a Identities h and size used in machine computation of case. sizes given with the adjusted h as listed. I
838
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